it's all rather brain-exploding, but to the limited extent I understand it the genes controlling skin colour exhibit incomplete dominance, the result of which if a darker-skinned and a lighter-skinned person mate is children that are always in between. They must be lighter-skinned than the dark-skinned parent and darker-skinned than the light-skinned parent, no other result is possible.
However if children of two such pairings mate all bets are off: a child could inherit the lighter-skin alleles from both parents and be lighter than either, the darker-skin and be darker than either, or a mix and be much the same. With incomplete dominance that appears to be the only genetic way to get children of strongly different colours to each other, though note that only the mixed-alleles would favour either parent in that respect.
Complete dominance could also lead to such results down the line, as recessive genes in both parents combine in a child to produce one of markedly different colour to either, and to siblings who received a dominant-recessive pairing or two of the same dominant. But I don't think there is a mechanism to produce the exact result specified of dark-skinned and light-skinned parents with no previous admixture producing a couple of kids of the one colour and a couple more of the other. And while the stripes might be very handsome I don't believe it could work that way; you would need specific genes evolved to produce different pigmentation in different areas of skin. Absent that you would always have a uniform colour, whether a blend or one colour or another.
Such at any rate is my limited understanding, derived from reading around on this interesting question. I put it out more to see if it's shot down than for any other reason. The only time I ever thought about the question before was to muse on what would happen if all gay people were born with blue skin. Pro: dating much easier, full minority recognition and rights much sooner. Con: 99.999% infanticide rate in most societies, most eras. I guess the con outweighs the two pros.